Networks of interdependent industrial assets (clients) are tightly coupled through physical processes and control inputs, raising a key question: how would the output of one client change if another client were operated differently? This is difficult to answer because client-specific data are high-dimensional and private, making centralization of raw data infeasible. Each client also maintains proprietary local models that cannot be modified. We propose a federated framework for causal representation learning in state-space systems that captures interdependencies among clients under these constraints. Each client maps high-dimensional observations into low-dimensional latent states that disentangle intrinsic dynamics from control-driven influences. A central server estimates the global state-transition and control structure. This enables decentralized counterfactual reasoning where clients predict how outputs would change under alternative control inputs at others while only exchanging compact latent states. We prove convergence to a centralized oracle and provide privacy guarantees. Our experiments demonstrate scalability, and accurate cross-client counterfactual inference on synthetic and real-world industrial control system datasets.
Figures & tables
Figure 1: A simplified view of the computation inside the client m
Figure 2: Global loss at the server and losses at the client from different models (augmented client loss: Lm,ak , proprietary client loss: T1∑t=1T∣∣rm,ct2∣∣2 are plotted against the number of iterations ( k ).
Figure 3: Disentanglement penalty at the server and δd at the clients vs. number of iterations.
Figure 4: Prediction error on the validation data for different models and the centralized oracle. ACM: Augmented Client Model, PCM: Proprietary Client Model
Figure 5: Comparison between Penalty Method (PM) and Augmented Lagrangian (AL) for the two client system. Ls , D , Local loss (client residual) Lm,a and δdm vs Number of iterations ( k ) are plotted. Note that we defined δdm:=∥ϕm−T1∑t=1T(∑n=mB^mnunt−1)∥2
Dm=16
Dm=32
Dm=64
Dm=128
Ls
D
Ls
D
Ls
D
Ls
D
0.7649
0.0034
1.0987
0.0041
1.4243
0.0046
1.1805
0.0047
Table 1: Ls and D by scaling measurement dim. Dm
M=2
M=4
M=8
M=16
Ls
D
Ls
D
Ls
D
Ls
D
0.0744
0.0013
0.0412
0.0010
0.1714
0.0036
0.3825
0.0069
Table 2: Server loss Ls by scaling number of clients M
Causal representation learning seeks latent features that reveal underlying causal variables and mechanisms for robust prediction, intervention, and transfer.
Root cause analysis (RCA) in networked industrial systems, such as supply chains and power networks, is notoriously difficult due to unknown and dynamically evolving interdependencies among geographically distributed clients. These clients represent heterogeneous physical processes and industrial assets equipped with sensors that generate large volumes of nonlinear, high-dimensional, and heterogeneous IoT data. Classical RCA methods require partial or full knowledge of the system's dependency graph, which is rarely available in these complex networks. While federated learning (FL) offers a natural framework for decentralized settings, most existing FL methods assume homogeneous feature spaces and retrainable client models. These assumptions are not compatible with our problem setting. Different clients have different data features and often run fixed, proprietary models that cannot be modified. This paper presents a federated cross-client interdependency learning methodology for feature-partitioned, nonlinear time-series data, without requiring access to raw sensor streams or modifying proprietary client models. Each proprietary local client model is augmented with a Machine Learning (ML) model that encodes cross-client interdependencies. These ML models are coordinated via a global server that enforces representation consistency while preserving privacy through calibrated differential privacy noise. RCA is performed using model residuals and anomaly flags. We establish theoretical convergence guarantees and validate our approach on extensive simulations and a real-world industrial cybersecurity dataset.
Ayush Mohanty, Paritosh Ramanan, Nagi Gebraeel
School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332 · School of Industrial Engineering and Management, Oklahoma State University, Stillwater, OK 74708
Networks of modern industrial systems are increasingly monitored by distributed sensors, where each system comprises multiple subsystems that generate high-dimensional time-series. These subsystems are often interdependent, making it important to understand how temporal patterns observed at one subsystem relate to those observed across others. This problem is particularly challenging in decentralized settings where raw measurements cannot be shared, and client observations are heterogeneous. Moreover, in practice, each subsystem (client) operates a fixed, proprietary model that cannot be modified or retrained, limiting the applicability of existing approaches. Separately, nonlinear dynamics make cross-client temporal interdependencies difficult to characterize, as they are embedded implicitly within nonlinear state-transition functions. This paper presents a vertical federated learning framework for learning one-step-ahead cross-client temporal interdependencies under these constraints. Each client maps high-dimensional local observations to low-dimensional latent states using a nonlinear state-space model, while a Graph Attention Network (GAT)-based central server learns a state-transition model over the communicated states. Given a candidate graph skeleton, the server learns state-dependent attention coefficients that capture cross-client interactions. We analytically relate the GAT attention coefficients to the Jacobian of the learned state-transition model, yielding a first-order characterization of cross-client interdependencies. We further establish bounded closeness of the learned dynamics and Jacobians to a corresponding centralized oracle. Synthetic experiments validate the theoretical results on controlled systems with known graph skeletons. Experiments on real-world datasets with unknown graph skeletons demonstrate superior federated root cause analysis over baselines.
While federated learning enables collaborative modelling on decentralised data, standard methods merely fit historical observations. This purely observational approach is fundamentally insufficient for interventional inference and policy evaluation, as sequential actions dynamically alter future states. We propose Fed-CausalDiff, a federated causal diffusion framework for do-simulation. The architecture decomposes the evolution of the latent state into a global causal score function and a local confounding score function. This design enables decoupled synchronisation (DSS), where clients aggregate only the shared causal mechanism while retaining site-specific confounders locally to handle heterogeneity. Experiments on four datasets demonstrate that Fed-CausalDiff achieves better ATE and policy-value estimation accuracy, offering a favorable trade-off between communication cost and inference fidelity.
Pengfei Li, Mohammad Khalil
Centre for the Science of Learning & Technology (SLATE), University of Bergen Bergen, Norway